Problems 1/19-23

18. Plot the radial momentum wave functions of the states above.

19. Use data table from Bethe and Salpeter to do this problem. If 1000 hydrogen atoms have been excited to the 6f state, how many photons at what wavelengths are you expected to detect?

If the excited atoms were created in flight moving at an energy of 100 keV, where do you put your photon detectors downstream?

20. On H_2 and H_2^+.

(a). What is the quilibrium distances for H2, for H2+?

(b) What is the energy difference between the ground state of H and of D?

(c) What is the vibrational frequency of H_2 at the ground level? At what proton impact energy can you use the Frank-Condon principle in describing the collision of protons on H_2?

(d) At room temperature what is the fraction of H_2 in the first vibrational state?

21. On the lifetime of some "famous" metastable states. Find out the lifetime of 2s state of atomic hydrogen. The 1s2s ^1S and ^3S states of He. Identify the radiative mode(s).

22. A helium atom has the configuration 1s2p, write down the FULL wave function including the radial, angular and spin functions. Consider both the singlet and the triplet states. Write down the expression which gives the singlet and triplet splitting. Which state is higher in energy in general?

23. If the helium atom has the 2p2p configuration, what are the possible terms, i.e., the possible L and S? According to the Hund's rule (what is that?), what are the energy ordering in this case?

Problems 2/2-6

24. This problem has to do with the spontaneous emission rate. Check with the equations in Section 4.6 of Bransden and Joachian, Physics of atoms and molecules. pp180-3. (this reference will be called BJ)

(a)From eq. [4.119] write down the numerical expression which relates the transition rates and the oscillator strength, if the energy separation is given in terms of atomic units. Check the correctness of your result by comparing the oscillator strength and the 2p lifetime given in Tables 4.1 and 4.2.
(b) From Bethe and Salpeter, find the transition rates to 1s for 2p,3p,4p 5p. Multiply each rate by n^3 and check if the scaled rate is nearly constant with respect to n. Consider atomic hydrogen
(c) do the same as in (b) for transitions from 3d,4d,5d,6d to 2p. Compared the scaled rate with the result in (b).
(d) Consider the scaling of the transition rates with respect to Z, what will be the lifetime of the 2p state of He+ ion?
(e) This part is a little bit more difficult. The equations used above are for the averaged rates. The averaged rate is defined as in eq. [4.118] of BJ which is an averaged over a statistical distribution of the initial magnetic states and summed over the final magnetic states. Suppose now experimentally you can populate a specific magnetic state, how does the emission rate vary with m? Work out an example such as the transition from 3d to 2p. For 3d, consider m=2 and m=1 cases. Hint: use the Winger-Eckart theorem. The 3-j symbols can be obtained using Mathematica.
(f) The reverse of (e) is used for optical pumping. Consider a 2p state which is initially statistically populated. If a linearly polarized light is used to excite 2p to 3d which then decays back to 2p, explain how the process will result in a selected populated magnetic states for 2p. Which m will be populated mostly?

25. Write down the hydrogen atom wave functions in the following approximations. (a) If there is no spin for the electron and the electron is in a 2p state.
(b) how many 2p states? That is, what are the degenerate states?
(c) If the electron has the spin, but there is no spin-orbit interaction, how many degenerate states are there? Write down the wave functions and identify the operators that give the quantum numbers used in your wave function.
(d) If there is spin-orbit interaction, what are the possible j's you can have? Write down explicitly the wave function for j=1/2 and m_j= 1/2. Use \alpha and \beta to denote spin-up and spin-down, respectively.
(e) Look up in some books and write down the Dirac wave function for the state given above.

26. Check with your q.m. books and write down

(a) an outgoing Coulomb wave function
(b) an ingoing Coulomb wave function

27. This exercise will take you through the basic calculations for He atom.

(a) write down the Hamiltonian for He atom including only the Coulomb interactions.
(b) We will treat the electron-electron interaction as perturbation. Write down the zeroth order wave function for the 1s(1)2p(2) state, meaning that the first electron is in 1s and the second is in 2p. Include all the degenerate states. Do not forget the spin. You should have six degenerate states.
(c) From the six degenerate states above, construct new eigenstates that are now the eigenstates of the zeroth order Hamiltonian, the total orbital and spin angular momentum operators and their projections on a space-fixed z axis. Also make sure that the resulting functions are totally antisymmetric. Identify the singlet and triplet states. With respect to the zeroth order Hamiltonian all these states are still degenerate. We are just making a unitary transformation among the degenerate states.
(d) Now use the first order perturbation theory to include electron-electron interaction. Show that this gives you a difference in singlet and triplet energy splitting.

Problems 2/9-13

29. (a) Sketch the elastic scattering cross sections for electrons colliding with atomic hydrogen in the energy region near 10 eV. Data can be found in G. Schulz, Rev. Mod. Phys. 45, 378 (1973). Some of the figures are in my lecture notes from 1997 Spring.
(b) Check what are the resonances that have been identified. Give the energies and the widths of the resonances Estimate the lifetimes of these states in seconds.
(c) Note that there is a "shape resonance" above the n=2 excitation threshold. What is its energy?

30. Describe how a resonance is parametrized using the Fano's theory. Can the same parameters describe a shape resonance?

31. Use all the approximation you know to estimate the total energies of the following doubly excited states for F$^{7+}$.
(a) nsnp for n=2,3,4,5,6
(b) 4snp for n=6,7,8
Also calculate the energies of F$^{8+}$ for n=3,4,5.
Express your answers in eV's.

A more complete prescription for evaluating energies for doubly excited states can be found in C. D. Lin and S. Watanabe, Phys. Rev. A35, 4499 (1987).

32. We will use the different classical models to estimate the electron capture cross section in this exercise for C4+ on H collisions.

(a) Use the overbarrier model to calculate the constant cross section;
(b) Use the Lagevin model to calculate the capture cross section at low energies;
(c) Use the Bohr-Linhard model to calculate the cross section at the high energies. Make sure that you went through the derivation of these formulae once yourself. Please cover the energy range of 10 meV/u to 1 MeV/u.
(d) Use the overbarrier model to estimate the principal quantum number n where the electron will be captured to.
(e) Look up the binding energies of C$^{3+}$(3s,3p,3d). Use the reaction window model to predict the dominant capture channels for collision energies at 0.4keV/u and at 20 keV/u.
(f) Look up this paper: " Low-energy electron capture by C4+ ions from atomic hydrogen" by F. W. Bliek, R. Hoekstra, M. E. Bannister, C. C. Havener pp. 426-431 Phys. Rev. A56 (1997) and compare with the measured data given here or from other sources quoted. Please also check the experimental method used, see C. Havener et al, Phys. Rev. A39, 1725 (1989).

33. If a doubly excited state is described by 2s4p$^1P^o$, what are the dominant final states after autoionization or radiative decay processes? Estimate the photon energy and the ejected electron energy assuming the two-electron ion has charge Z=6.
If the initial state is given by 2p2p $^3P^e$, this state cannot autoionize, why? Can 3p3p$^3P^e$ state autoionize?

34. A hydrogen atom in the n=3 state has many degenerate states.
(a) A weak electric field of 100 v/cm is applied to the hydrogen atom, what is this field strength in terms of atomic units?
(b) Let the quantization axis be along the direction of the field, calculate the shift of the levels (for m=0,1 and 2) in eV's. Consider linear Stark effect only.
(c) Estimate the field strength that you begin need to worry about that the linear Stark effect is not valid.
(d) Estimate the lifetime of the lowest n=3 state in the field of 100 V/cm.

35. Consider a lithium-like carbon ion C$^{3+}$. The three electrons have the configuration 1s2s2p. Give the possible L, S and J and identify the dominant radiative and auger decay channels and the final states.
Note: consider each J separately and assess which state is metastable.